Effects of Additives in Solution Crystallization Dissertation zur Erlangung des akademischen Grades Doktor−Ingenieur (Dr.−Ing.) genehmigt durch die Mathematisch−Naturwissenschaftlich−Technische Fakultät (Ingenieurwissenschafticher Bereich) der Martin−Luther−Universität Halle−Wittenberg von Herrn M.Sc. Eng. Sattar Al−Jibbouri geboren am 29.12.1972 in Kadisia/Irak Dekan der Fakultät: Prof. Dr. Dr. rer. nat. habil. H. Pöllmann Gutachter: 1. Prof. Dr. J. Ulrich 2. Prof. Dr. M. Pietzsch 3. Prof. Dr. A. König Merseburg, 19.12.2002 urn:nbn:de:gbv:3-000004666 [http://nbn-resolving.de/urn/resolver.pl?urn=nbn%3Ade%3Agbv%3A3-000004666]
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Effects of Additives in Solution Crystallization · crystallization kinetics, most probably, is due to the impurity adsorption on the crystal surface. Therefore, an understanding
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My grateful appreciation to my supervisor, Prof. Dr-Ing. habil.
Joachim Ulrich, for his helpful, guidance and a continued encouragement
throughout this work.
Also deep thanks are to examining committee Prof. Dr. Roggendorf,
Prof. Dr. König, Prof. Dr. Pietzsch, Prof. Dr. Lempe, Prof. Dr. Kressler, Prof.
Dr. Leps and PD Dr. Brendler.
My deep thanks are extended to the Martin-Luther-Universität
Halle-Wittenberg for their support of this project which led to this work.
My deep thanks for all my colleagues, (Dr. Mohring, Dr. Wanko, Tero,
Junjun, Christine, Stefan, Torsten, Bernd, Kim, Dannail, Aiman, Mirko,
Mandy, Peter, Andrea and Uta), and especially (Mrs. Dr. Heike Glade, Mrs.
Cornelia Lorentz and Mrs. Ing. Frauke Mätsch) for their help during my
studying.
Sattar Al-Jibbouri
Table of Contents
1. Introduction 1
2. State of the art 3
2.1. Kinetic effects (crystal side) 3
2.2. Thermodynamic effects (solution side) 5
2.3. The aim of the present work 7
3. Theory 8
3.1. The Three-Step-model 9
3.2. The concept of effectiveness factors 13
3.3. Model for crystal growth in the presence of impurities 15
3.4. Electrical double layer 19
3.4.1. Origins of surface charge 21
3.4.1.1. Ion dissolution 22
3.4.1.2. Ionization of surface groups 23
3.4.2. Electrophoresis 24
3.4.3. The diffusion double layer (The Gouy−Chapman model) 25
3.4.3.1. The Poisson-Boltzmann equation 26
3.4.3.2. The Grahame equation 31
3.4.3.3. The capacity of the double layer 32
3.4.4. Additional description of the electrical double layer 32
4. Experimental Work 34
4.1. Fluidized bed experiments 34
4.1.1. Fluidized bed measurement equipment 34
4.1.2. Procedure 34
4.2. Electrophoretic-mobility measurements 37
4.2.1. Procedure 38
5. Results and discussion 39
5.1. NaCl experiments 39
5.2. MgSO4•7H2O experiments 43
6. Interpretation of results 47
6.1. The magnitude of the two resistances (diffusion and integration
steps)
47
6.1.1. Nacl experiments 47
6.1.2. MgSO4•7H2O experiments 49
6.2. Kinetic effect 51
6.2.1. NaCl experiments 51
6.2.2. MgSO4•7H2O experiments 54
6.3 Thermodynamic effect 58
6.3.1. The effect of pH (MgSO4•7H2O experiments) 58
6.3.2. The effect of K + ions (MgSO4•7H2O experiments) 60
6.3.3. The effect of hydro-complex ions 61
6.3.3.1. MgSO4•7H2O experiments 61
6.3.3.2. NaCl experiments 65
6.4. Electrical double layer 67
6.4.1. Charged particles 67
6.4.2. Measuring crystal charge (ζ−potential) 68
6.4.3. Effect of pH 71
6.4.4. Effect of adsorption ions 74
6.5. Summary of results 80
7. Summary 82
8. Zusammenfassung 84
9. Notation 86
10. References 89
Abstract
In this study a fluidized bed crystallizer is employed to investigate the growth and
dissolution rate of MgSO4•7H2O and NaCl crystals. In the experiments the supersaturation, impurity concentration and pH-values in the solution were varied. The electrophoretic
mobility measurements by Laser-Doppler electrophoresis (ζ−potential measurements) are
reported for MgSO4•7H2O crystals. These measurements for inorganic salt have been made
for the first time and allow the surface charge to be predicted for MgSO4•7H2O crystals in
their saturated solution. Therefore, knowing the surface potential by measuring ζ−potential can help to explain the crystallization phenomena which are not clear up to now. In general,
the results show that the MgSO4•7H2O crystals have a positive ζ−potential charge. At low pH the surface will acquire more positive charge and at high pH a build up of negative charge will take place, hence, the crystal growth is suppressed. In this study it was proven
that the growth rates of MgSO4•7H2O crystals are suppressed by traces of Fe+2/Ni+2 ions.
surface potential, surface charge, zeta−potential. In dieser Studie wird ein Flüssigbettkristallisator eingesetzt, um die Wachstum- und
Auflösungrate MgSO4•7H2O und NaCl der Kristalle nachzuforschen. In den Experimenten wurden die Übersättigung, die Störstellenkonzentation und die pH-werte in der Lösung verändert. Die elektrophoretischen Mobilität Maße durch Laser-Doppler Elektrophorese
(Zeta Potential Maße) für MgSO4•7H2O Kristalle berichtet. Diese Maße für anorganisches Salz sind zum ersten Mal gebildet worden und erlauben, daß die Oberflächenaufladung für
MgSO4•7H2O Kristalle in ihrer gesättigten Lösung vorausgesagt wird. Folglich kann das Kennen des Oberflächenpotentials, indem es Zeta Potential mißt, helfen, die Kristallisation Phänomene zu erklären, die nicht zu jetzt aufräumen sollen. Im allgemeinen zeigen die
Resultate, daß die MgSO4•7H2O Kristalle eine positive Zeta Potential Aufladung haben. Bei niedrigem pH erwirbt die Oberfläche positivere Aufladung und bei hohem pH findet ein Aufbau der negativen Aufladung statt, folglich wird das Kristallwachstum unterdrückt.
In dieser Studie wurde es nachgewiesen, daß die Wachstumsraten der MgSO4•7H2O Kristalle durch Spuren der Ionen Fe+2/Ni+2 unterdrückt werden. Keywords: Anorganisches Salz, Wachstumsrate, Störstellen, Kinetische Effekte,
For many multicomponent systems the solubility data available are scarce and
frequently unreliable, thus becoming of little use for most engineering purposes.
Because entire phase diagrams cannot be obtained through experiment in a reasonable
length of time, several thermodynamic models have been proposed to predict the
solubility of salts in aqueous brines. Some of the most successful models make use of
the Pitzer equation to describe the thermodynamic properties of aqueous electrolytes,
over large ranges of ionic strength [50-52]. The Pitzer model also allows to extend these
calculations to elevated temperatures, if some empirical functions of temperature
obtained from binary and ternary data were included. Therefore, the solubility of
electrolyte solutions can be calculated from the thermodynamic considerations provided
that equilibrium constants and the activity coefficients can be obtained. The solubility of
the main salt may be changed by adding either another salt as impurities or a high
concentrations of acid or base (changing pH-value of the solution).
Consequently, the pH of the solution may affect not only the growth and
dissolution rate [53, 54], but also the different physical properties of the saturated
solution like osmotic pressure, the density, the surface tension and the metastable region
[55]. Only a limited number of studies deals with the effect of pH level on the
crystallization kinetics. Some studies considered the effect of the pH of a solution (water
as solvent) on the crystallization kinetics in the system, in which hydrogen ion is
incorporated in the equilibrium constant of the main salt that should be crystallized.
Examples in this category are the phosphate salts [56-58] and calcium carbonate [59,
60]. Other studies (Seifert [61], Langer [62]) investigated the effect of controlling the pH
of the solution, by the addition of an acid or base to the required solution, on the
crystallization kinetic. Seifert [61] investigated the influence of the pH on the growth
rate of KCl. He reported that no effect in the acidic medium on the replacement velocity
of the (100) face of single crystals, but an increase in the growth rate was found in the
alkaline medium. This increase in growth rate is dramatically pronounced by pH vales of
≥ 12. The different pH levels in the solution are achieved by addition of HCl and KOH.
Langer [62] studied the effect of pH on the growth rate of NaCl. His results show a
maximum crystal growth in the neutral solution, with lower crystal growth rates in both
acidic and alkaline solutions.
State of the art6
There are several more or less satisfactory explanations of the effect of pH on
crystallization. A plausible explanation tells that the presence of free acids or bases
modifies the nature and the concentration of ions in solution [12]. Mohameed and Ulrich
[54] explain the effect of pH on crystal growth in terms of a structure in a solution,
namely of a hydration of ions. Most cations and OH− ions are hydrated, the largest
hydration enthalpy has the H+ ion so that its presence in solution has a stronger tendency
of interaction with water molecules than, e.g., the K+ ion so that a competition of ions to
acquire water molecules takes place. The K+ ions have smaller chances to be fully
hydrated and therefore they tend to drift towards the crystal surface rather than to remain
in the solution. On the another hand, the OH− ions as a structure former has a stabilizing
effect on the solution so that K+ ions try to remain in the solution; nevertheless, at high
OH− concentrations, the tendency of hydration of the OH− ions prevails and the K+ ions
are pushed off from the solution again. Kuznetsov and Hodorowicz [63] presented a
hypothesis that thermal vibrations of ions in the solution excite electromagnetic waves
with the frequency of which depends, among others, on the concentration of the
hydroxonium ions. A correlation of the frequency with the vibrations of particles in the
crystal lattice can affect the crystallization rate.
Recently there has been more attention paid to the effect of the pH value on
crystallization kinetics, especially in combination with other additives e.g. Kubota et al.
[64] studied the effect of four chromium (III) salts as impurities on the growth and
dissolution rates of K2SO4 over a wide range of pH. In this study, it was reported that the
adsorption of H+ ions at the crystal surface is not possible and if that would be the case,
the dissolution rate would be lower. Baes and Mesmer [65] proved that in the case of
presence of some chrome complexes in aqueous solutions, the degree of change of the
pH level is the key factor by which impurity affects the crystallization process. The
experimental results reported by Mullin et al. [57] show that the pH level of the solution
in the presence of Cr+3, Fe+3, Al+3 affects the habit of ADP and KDP single crystals.
Takaski et al. [60] studied the effect of the presence of ferrous ions as hydroxide on the
growth rate of calcium carbonate. It was found that in the range of pH between 7 and 8
there is no effect on the growth rate.
The effect of hydronium ions on the potential of the diffusion layer was not
considered in the previous studies. The dissociation of a proton from water molecules
may effect the structure of the solution and the potential of the crystal/solution interface.
State of the art7
If it is assumed that the protons will be adsorbed at the surface of the crystal, the
crystals will be changed and this leads to the development of the so called electrical
double layer (see Ch. 3) between the counterions in the solution near the surface and the
charge on the surface.
2.3.2.3.2.3.2.3. The aim of the present work The aim of the present work The aim of the present work The aim of the present work
The growth of a single seed crystal differs from the environment in an industrial
crystallizer where many crystals are growing in a suspension. The most common method
to obtain crystal growth kinetics involving suspension involves the use of a stirred tank
crystallizer (MSMPR) or a fluidized bed crystallizer. Therefore, the aim of the present
work is:
• To evaluate the relative magnitude of the two resistances in series, diffusion and
surface integration by curve fitting of the growth data.
• To show that the kinetic models for crystal growth in the presence of impurities
adsorbing at kinks and surface terraces on the F faces of single crystals are also valid
for suspension growth condition as in industrial crystallizers.
• To propose a physical explanation for the influence of univalent or divalent positive
ions (cations) on the structure of the solution based on the thermodynamic properties
of the saturated solution. (i.e. the effect of impurities can be explained by a
mechanism in which the hydrolysis product (hydro-complex) is adsorbed on the
growth layer of crystal surface and retard growth).
The major objective of the present investigation is to propose a new explanation
for the effect of impurities or changing the pH value of the solution on the crystallization
kinetics, based on the hypothesis, that the crystal growth rate of salts is dominated by
the surface potential distribution. Specific cation/anion adsorption is the main reason
for the occurrence of changing on the surface charge and this adsorption is a function of
the surface quality. The adsorption of cation/anion ions on the crystal surface has a very
strong effect on the electrical double layer. Consequently, they should have a specific
effect on crystal growth, depending on how the surface charge is it will affect in same
way (increasing /decreasing or reversing) the sign of the charge. This means that any
electrical potential on the crystal surface may lead to an increase or decrease in the
crystal growth rate, or have no effect on it, depending on the dominant effect.
Theory8
3.3.3.3. TheoryTheoryTheoryTheory
Crystallization is a separation and purification technique employed to produce a
wide variety of materials. Crystallization may be defined as a phase change in which a
crystalline product is obtained from a solution. A solution is a mixture of two or more
species that form a homogenous single phase. Solutions are normally thought of in terms
of liquids, however, solutions may include solids suspension. Typically, the term
solution has come to mean a liquid solution consisting a solvent, which is a liquid, and a
solute, which is a solid, at the conditions of interest. The solution to be ready for
crystallization must be supersaturated. A solution in which the solute concentration
exceeds the equilibrium (saturated) solute concentration at a given temperature is known
as a supersaturated solution [66]. There are four main methods to generate
supersaturation that are the following:
• Temperature change (mainly cooling),
• Evaporation of solvent,
• Chemical reaction, and
• Changing the solvent composition (e.g. salting out).
The Ostwald-Miers diagram shown in Fig. 3.1. illustrates the basis of all
methods of solution growth. The solid line represents a section of the curve for the solute
/ solvent system. The upper dashed line is referred to as the super-solubility line and
denotes the temperatures and concentration where spontaneous nucleation occurs [67].
The diagram can be evaluated on the basis of three zones:
• The stable (unsaturated) zone where crystallization is impossible,
• The metastable (supersaturated) zone where spontaneous nucleation is
improbable but a crystal located in this zone will grow and
• The unstable or labile (supersaturated) zone where spontaneous nucleation is
probable and so the growth.
Crystallization from solution can be thought of as a two step process. The first
step is the phase separation, (or birth), of a new crystals. The second is the growth of
these crystals to larger size. These two processes are known as nucleation and crystal
growth, respectively. Analysis of industrial crystallization processes requires knowledge
of both nucleation and crystal growth.
Theory9
The birth of a new crystals, which is called nucleation, refers to the beginning of
the phase separation process. The solute molecules have formed the smallest sized
particles possible under the conditions present. The next stage of the crystallization
process is for these nuclei to grow larger by the addition of solute molecules from the
supersaturated solution. This part of the crystallization process is known as crystal
growth. Crystal growth, along with nucleation, controls the final particle size distribution
obtained in the system. In addition, the conditions and rate of crystal growth have a
significant impact on the product purity and the crystal habit. An understanding of the
crystal growth theory and experimental techniques for examining crystal growth from
solution are important and very useful in the development of industrial crystallization
processes. The many proposed mechanisms of crystal growth may broadly be discussed
under a few general headings [67-70]:
• Surface energy theories
• Adsorption layer theories
• Kinematic theories
• Diffusion - reaction theories
• Birth and spread models
Figure 3.1.: Ostwald-Miers diagram for a solute/solvent system [67].
3.1.3.1.3.1.3.1. The Three-Step-model The Three-Step-model The Three-Step-model The Three-Step-model
Modelling of crystal growth in solution crystallization is often done by the Two-
Step-Model. The Two-Step-model describes the crystal growth as a superposition of two
resistances: bulk diffusion through the mass transfer boundary layer, i.e. diffusion step,
Stable
Labile
Metastable
Temperature [°°°°C]
Con
cent
ratio
n [g
/100
g H2O
]
Theory10
and incorporation of growth unites into the crystal lattice, i.e. integration step [67, 68].
The overall growth rate is expressed as:
RG = kd (Cb− Ci) (diffusion step), (3.1)
RG = kr (Ci− C*)r (integration step), (3.2)
RG = kG g*b )C(C − (overall growth), (3.3)
where (Cb − C*) is the supersaturation.
The Two-Step-Model is totally ignoring the effect of heat transfer on the crystal
growth kinetics. In the literature there is little evidence for the effects of heat transfer on
the crystal growth kinetics in the case of crystallization from solution. Matsuoka and
Garside [3] give an approach describing the combined heat and mass transfer in crystal
growth processes. The so called Three-Step-model of combined mass and heat transfer
takes the above mentioned effects into account [1-3]. A mass transfer coefficient is
defined which includes a dimensionless temperature increment at the phase boundary
constituted by the temperature effect of the liberated crystallization heat and the
convective heat transfer. For simplicity the transport processes occurring during growth
will be described in terms of the simple film theory. This has the advantage that the
resulting equations can be easily solved and the predictions do not differ significantly
from those derived using the boundary layer theory [71, 72]. Conditions in the fluid
adjacent to the growing crystal surface are illustrated in Fig. 3.1.1.. The mass transfer
step can be presented by the equation:
RG = kd (Cb−Ci)= kd [(Cb−C*b) − (Ci− C*
i) − (C*i − C*
b)]
= kd [∆Cb−∆Ci − (Ti−Tb) dTdC *
] (3.4)
where C*i and C*
b are the saturation concentrations evaluated at the interface and bulk
temperatures, respectively. The effect of bulk flow, important at high mass fluxes, is
neglected in Eq. 3.4. It is also assumed that the temperature difference (Ti−Tb) is
sufficiently small for the solubility curve to be assumed linear over this temperature
range. A heat balance relating heat evolution to convective transfer gives:
Theory11
RG ΔH
)bTih(T
−
−= (heat transfer), (3.5)
Combination of Eqs. 3.4 and 3.5 gives
RG
dT*dC
hdkΔH
1
dk
⋅×−
+= (∆Cb−∆Ci)
d1dkβ+= [∆Cb−∆Ci)= kd´(∆Cb−∆Cb) (3.6)
βd = dT*dC
hdkΔH⋅
×− (3.7)
Where βd is defined by Matsuoka and Garside [3] as a dimensionless number for the
temperature increase at the crystal surface and therefore as measure of the heat effect on
growth kinetics.
Figure 3.1.1.: Concentration and temperature profiles to
in the simple film theory [1].
The analogy between mass transfer and heat transfer is giv
2/3Le cp cρ2/3
PrSc cp cρ
dkh ≡
=
substitution Eq. 3.8 into Eq. 3.7 gives the following equat
βd = dT*dC
2/3Le cp cρΔH ⋅−
Crystal
Driving forcefor diffusion
Driving forcefor reaction
Adsorption layer
Cb
C*b
C*i
TiTb
Ci
δC
δT
Concentrationor
Temperature
the crystal surface as assumed
en by [73]:
(3.8)
ion:
(3.9)
Theory12
The general expression for the overall growth rate can be obtained by combining Eqs.
3.1, 3.2 and 3.6:
RG = kr r
)dk
dβ1GR)*Cb(C
+−− (3.10)
Matsuoka and Garside [3] give a limit βd must be > 10 −2, for values below which the
influence of the heat transfer on the crystal growth kinetics can be neglected.
The dissolution process is, on the contrary, quite frequently described only by use
of the diffusion step. What is not true since there is definitely a surface disintegration
step [4, 5]. In other words dissolution is the 100 % opposite of crystal growth. However,
a justification for the model assumption that dissolution can be seen as just diffusion
controlled is due to experimental results which show a linear dependence on the
concentration difference (undersaturation). Furthermore, the dissolution process is
happening according to literature much faster (4 to 6 times) than the crystal growth
process so that a possible surface reaction resistance is here difficult to observe [4, 5].
The assumption that the dissolution of crystals involves the sole diffusion step is
therefore, in many case valid:
RD = kd (C* − Cb) (3.11)
Two methods, the differential and integration method, are mainly used for the
measurements of the growth rates in fluidized bed experiments [74]. In this study the
differential method was used. In the differential method, the crystallization is seeded by
adding a few grams of crystals with a known sieve aperture into a supersaturated
solution. The seed crystals grow in the supersaturated solution. Since the amount of
crystals is small, it is assumed that the concentration of the solution does not change
during the growth. The other assumptions are as follows:
• The number of seed crystals put into the crystallizer is equal to the number of
crystals taken out from the crystallizer.
• There is no crystal loss, an assumption which is always valid for an experienced
experiment.
• The shape factor of the growing crystals are considered to be the same. This
assumption is not always true especially in the case of surface nucleation. In this
case, growth values are thought of as average values.
Theory13
If the amount of the crystals put into the crystallizer is M1 and the amount of the
crystals taken out from the crystallizer is M2, they can be related to the size of the
crystals L as shown in the following equations [75]:
M1 = αρ 31L , (3.12)
M2 = αρ 32L , (3.13)
where L1 and L2 are the characteristic size of the crystals input and the output,
respectively. The overall linear growth rate G (m/s) is defined as the rate of change of
characteristic size:
G = tΔL (3.14)
The expression for the growth rate in terms of size of the seed crystals and the weight of
the crystals can be given by:
G =
−
1
MM
tL
1/3
1
21 (3.15)
G and RG are related to each other as follows:
RG = 1
1
β3α ρc G (3.16)
where β1 and α1 are surface and volume shape factors, respectively. M1 and M2 are
experimentally obtained. The growth rate, RG, and the dissolution rate, RD, are calculated
from Eq. 3.16 by knowing L1 and t.
3.2.3.2.3.2.3.2. The concept of effectiveness factors The concept of effectiveness factors The concept of effectiveness factors The concept of effectiveness factors
When crystals grow the rate at which solute is deposited in the crystal lattice is
controlled by two resistances in series, those offered by diffusion through the boundary
layer and by reaction at the crystal surface. If the rate equations for these two steps are
known, the overall crystal growth rate can be easily calculated. It is much more difficult
to deduce the kinetics of the individual resistances from measured overall growth rates.
Therefore, a quantitative measure of the degree of diffusion or surface integration
control may be made through the concept of effectiveness factors. A crystal growth rate
effectiveness factors, η, is defined by Garside [76] as the ratio of the overall growth rate
to the growth rate that would be obtained if diffusion offered negligible resistance is
given by:
Theory14
ηr = (1 − ηr Da)r (3.17)
where ηr is the integration effectiveness factor and Da is the Damköhler number for
crystal growth which represents the ratio of the pseudo first order rate coefficient at the
bulk conditions to the mass transfer coefficient, defined by:
Da = dkrk (Cb − C*)r-1 (3.18)
It will also be convenient to define a diffusion effectiveness factor, ηd as:
ηd = Da (1 − ηd)r (3.19)
The heat of crystallization produced at the crystal surface will change the
solution temperature at this point and hence alter the rates of the kinetics processes.
Consequently the effectiveness factor will change from that evaluated under bulk
conditions. The non-isothermal effectiveness factor, η´, is defined as the ratio of actual
growth rate to the rate that would be obtained if the bulk liquid conditions assumed to
exist at the crystal surface:
η´ = )conditions bulk (i.e. bT and bΔC at rate growth
)conditions interface (i.e. iT and iΔC at rate growth (3.20)
an analysis similar to that of Carberry and Kulkrani [72] for chemical reaction can be
applied to the crystal growth case to yeild
η´ = Da (1−η´) r exp
−
++− − 1
)1(´11
10daD ββη
ε (3.21)
where the Damköhler number crystal growth, Da, is defined by
Da = ´dkb r,k
(∆Cb )r-1 (3.22)
and represent the ratio of the pseudo-first order rate coefficient at the bulk conditions to
the mass transfer coefficient. The Arrhenius number is defined by:
ε0 = E/RTb (3.23)
and
Theory15
β = bTbΔC
2/3Le cp cρΔH ⋅− (3.24)
is the ratio of the interface adiabatic temperature rise to the bulk temperature. When
βd<<1, Eq. 3.21 becomes identical to that given by Carberry and Kulkrani [72], i.e.:
η´ = Da (1−η´) r exp
−
+− 1
´11
0 βηε
aD (3.25)
3.3.3.3.3.3.3.3. Model for crystal growth in the presence of impurities Model for crystal growth in the presence of impurities Model for crystal growth in the presence of impurities Model for crystal growth in the presence of impurities
It is well known that the influence of impurities on the crystal form and the
growth rate is based on the adsorption of the foreign molecules on the surface. The
change of crystal form is based on a difference in adsorption energies on different crystal
faces. Foreign molecules will be adsorbed preferentially on surfaces where the free
adsorption energy has its maximum. Surface adsorbed impurities can reduce the growth
rate of crystals by reducing or hindering the movement of growth steps. Depending on
the amount and strength of adsorption, the effect on crystal growth can be very strong or
hardly noticeable.
The step advancement velocity is assumed [49] to be hindered by impurity
species adsorbing on the step lines at kink sites by a modified mechanism, the original
version of which was proposed by Cabrera and Vermileya [14]. Step displacement is
pinned (or stopped) by impurities at the points of their adsorption and the step is forced
to curve as shown schematically in Fig. 3.3.1..
The advancement velocity of a curved step, vr, decreases as the radius of
curvature, ρ, is reduced and it becomes zero just at a critical size, r =rc. It is given
simply by the following equation [78], if the relative supersaturation is small (σ << 1):
0νrν =
rcr1− , (3.26)
where v0 is the velocity of linear step and rc is the critical radius of a two-dimensional
nucleus. At r ≤ rc the step cannot move. The instantaneous step advancement velocity
changes with time during the step squeezes out between the adjacent adsorbed impurities
because the curvature changes with time. The maximum velocity is v0 (of a linear step)
Theory16
and the minimum instantaneous step velocity vmin is given at a curvature of r = l/2 (l is
average spacing between the adjacent adsorbed impurities) by:
0νminν
= (l/2)
cr1− . (3.27)
Time-averaged advancement velocity v of a step is approximated by the arithmetic mean
of v0 and vmin [77] as:
v = (v0 + vmin)/2 . (3.28)
Figure 3.3.1.: Model of impurity adsorption. Impurity species are assumed to be
adsorbed on the step lines at kink sites and to retard the advancement of
the steps [77].
Combining Eqs. 3.27 and 3.28 one obtains the following equation for the average step
advancement velocity as a function of the average spacing between the impurities, l:
0νν =
lcr1− . (3.29)
while v = 0 for l ≤ rc.
This simple equation was thus obtained by assuming the linear array of sites on the step
lines and by using the arithmetic mean of the maximum and minimum step velocities as
an average step velocity.
Theory17
The coverage of active sites by impurities θ can be related to the average distance
between the active sites λ, from a simple geometric consideration, under the assumption
of linear adsorption on the step lines (linear array) as:
θ = λ/l (3.30)
on the other hand, the critical radius of a two-dimensional nucleus is given by Burton et
al. [78] as:
rc =
Tσka γ
B (for σ << 1) (3.31)
Insertion of Eqs. 3.30 and 3.31 into Eq. 3.29 gives the following equation:
0νν =1 −
λTσka γ
B
θ, (3.32)
where γ is the linear edge free energy of the step, a is the size of the growth unit (area
per growth unit appearing on the crysatl surface), kB is the Boltzmann constant, T is the
temperature in Kelvin.
As soon as kinks and steps are occupied by foreign molecules, the coverage of crystal
faces causes a reduction in growth rate [48]. If all active centres for growth are blocked,
growth rates can be reduced to zero. Kubota et al. [77], introduce the impurity
effectiveness factor, α. The effectiveness factor α is a parameter accounting for the
effectiveness of an impurity under a given growth condition (temperature and
supersaturation). Thus, the step advancement velocity can be written as a function of
temperature and supersaturation:
α =
λTσka γ
B (3.33)
Eq. 3.32 can be changed to
0νν =1 − αθ, (for αθ < 1) (3.34)
where v = 0 for αθ ≥ 1.
This impurity effectiveness factor can be less than or equal or greater than one. α
decreases with increasing supersaturation and is independent of K. In Fig. 3.3.2., the
Theory18
relative step velocities, calculated from Eq. 3.34, are shown for different effectiveness
factors, α, as a function of the dimensionless impurity concentration Kcimp.
Figure 3.3.2.: Theoretical relationship
and the dimensionless impuri
(diagram based on the work of
It is clear from Fig. 3.3.2. that, w
steeply with increasing impurity conce
Kcimp. For α = 1, a full coverage of the
zero. For α < 1, however, the step veloci
zero value as Kcimp is increased. This va
value of α = 0.
If an equilibrium adsorption is
coverage θ in Eq. 3.36 is replaced by an
0νν =1 − αθeq
and the step advancement velocity may
cimp if an appropriate isotherm is emplo
used for this purpose. Therefore, the cov
by the usual adsorption isotherms [35, 79
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15
Dimensionless impurity
Rel
ativ
e m
ass g
row
th r
ate
RG
/RG
0 [-]
α < 1 (weak impurity)
b
ty
[
h
n
ty
lu
as
e
b
y
e
,
c
α = 1 (weak impurity)
α > 1 (strong impurity)
α = 0
etween the relative mass growth rate, RG/RGo,
concentration, Kc, for different value of α.
77]).
en α > 1, the relative velocity decreases very
tration and reaches zero at a small value of
crystal surface leads to step velocity equal to
never approaches zero but approaches a non-
e is increasing with smaller α and is one at a
sumed for an impurity [49, 77], the surface
quilibrium value θeq:
(3.35)
e related to the concentration of the impurity
ed. Although any adsorption isotherm can be
rage, θeq, of adsorption sites may be described
80]:
20 25 30
oncentration Kc [-]
Theory19
θ eq= imp
imp
Kc1Kc+
(Langmuir isotherm) (3.36)
In this equation K is constant. The constant K of Eq. 3.36 is given by [79, 80]:
K = exp(Qdiff /R T) (3.37)
where Qdiff is the differential heat of adsorption corresponding to θeq.
In the case of a spiral growth mechanism, the relationship between the step
velocity at a crystal face, ν, and the fraction coverage, θeq, of the surface may be given
by [81, 82]:
(ν−ν/νo) n = αnθeq (3.38)
The exponent n=1 and 2 represents the case at which impurity adsorption occurs at kinks
in step edges and on the surface terrace, respectively.
The relative step velocity in Eq. 3.38, can be replaced by the relative growth rate
RG/RGo if the growth rate is assumed to be proportional to the step velocity:
(RGo− RG / RGo) n = αnθ eq (3.39)
The previous model of impurity adsorption considering kinks and the surface terraces
deal with the kinetic aspect of adsorption of impurities of F faces, neglecting the
thermodynamic effects. Therefore, for all the above equations it is true that growth rates
are reduced, when impurities are present in the solution. Generally, experiments carried
out in a fluidized bed crystallizers [83-86] showed that the addition of small amounts of
impurities lead to a decrease in growth rates. This is in good agreement with theoretical
Electrophoresis is defined as the migration of ions under the influnce of an
electric field. The force (F = qE) imparted by the electrical field is proportional to its
effective charge, q, and the electric field strength, E. The translation movement of the
ion is opposed by a retarding frictional force (Ff = fv), which is proportional to the
velocity of the ion, v, and the friction coefficient, f. the ion almost instantly reaches a
steady state velocity where the acceleration force equals the frictional force.
qE = f v ⇒ v = (q/f) E = u E (3.47)
Here u is the electrophoretic mobility of the ion, which is a constant of proportionality
between the velocity of the ion and the electric field strength. The electrophoretic
mobility is proportional to the charge of the ion and inversely proportional to the friction
coefficient. The friction coefficient of the moving ion is related to the hydrodynamic
radius, a1, of the ion and the viscocity, µ, of the surrounding medium, f = 6πµa1, because
u = q/f, a larger hydrodynamic radius translates to a lower electrophoretic mobility.
The effective charge arises from both the actual surface charge and also the charge in the
double layer. The thickness of the double layer is quantified by a1 parameter with the
dimensions of inverse length k, so that the dimensionless number ka1 effectively
measures the ratio of particle radius to double layer thickness. The figure below
illustrates the typical situation.
Figure 3.4.1.: Apparent charge distir
potential [88].
k
+
+
+
++
+
a1
−
ibu
−+
++
−
− −
−
−
1/−
−
−
−
−− −
−
−
tion
−
−
−
−
−
−
−
−
−
−
−
−
−+
around a spherical particale at low
Theory25
It turns out that q can be estimated using some approximations. Providing that
the value of charge is low, (zeta potential less than 30 mV or so) the Henry equation can
be applied [88, 98-100]:
u = (2εζ / 3µ) f(ka1) (3.48)
Henry‘s function, f(ka1), vaires smoothly from 1 to 1.5 as ka vaires from 0 to ∞, these
corresponding to limiting cases where the particle is much smaller than the double layer
thickness, or much larger.
3.4.3.3.4.3.3.4.3.3.4.3. The diffusion double layer (The The diffusion double layer (The The diffusion double layer (The The diffusion double layer (The GouyGouyGouyGouy−−−−Chapman model)Chapman model)Chapman model)Chapman model)
Surface charge cause an electrical field. This electrical field attracts counter ions.
The layer of surface charges and counter ions is called electrical double layer. The first
theory for the description of electrical double layers comes from Helmholtz started with
the fact that a layer of counter ions binds to the surface charges [88-96]. The counter
ions are directly adsorbed to the surface. The charge of the counter ions exactly
compensates the surface charge. The electrical field generated by the surface charge is
accordingly limited to the thickness of a molecular layer. Helmholtz could interpret
measurements of the capacity of double layers; electrokinetic experiments, however,
contradicted his theory.
Gouy and Chapman went a step further. They considered a possible thermal
motion of the counter ions. This thermal motion leads to the formation of a diffuse layer,
which is more extended than a molecular layer [88]. For the one-dimensional case of a
planar, negatively charged plane this is shown in the illustration. Gouy and Chapman
applied their theory on the electrical double layer of planar surface. Later, Debye and
Hückel calculated the behaviour around spherical solids.
Fig. 3.4.2. portrays schematically the discrete regions into which the inner part of
the double layer has been divided. First there is the layer of dehydrated ions (i.e. Inner
Helmholtz Plane, I.H.P.) having potential, Ψ0, and surface charge, σ0, and second there
is the first layer of hydrated ions (i.e. Outer Helmholtz Plane, O.H.P.) having potential,
Ψd, and charge, σd. The O.H.P. marks beginnings of the diffuse layer [88, 92].
Theory26
Figure 3.4.2.: Schematic
3.4.3.1.3.4.3.1.3.4.3.1.3.4.3.1. The Poisso The Poisson The Poisso The Poisson
The aim is to ca
Therefore, here a plane i
density, ρ, which is in co
related by the Poisson equ
xΨ2
2xΨ2
2xΨ2Ψ2
∂
∂+∂
∂+∂
∂=∇
With the Poisson equation
of all charges are known.
free to move. Since their
unknown, the potential
Additional information is
This additional for
solution from far away cl
ion density would be:
T/kiWe inin B0 −
=
ni0 is the density of the ith
ion concentration depen
example, if the potential
there will be more anions,
I.H.P Ψ0, σ0O.H.P Ψd, σd
ζ-potential
Primary bond
Diffused layer
−
+
Hydrated cationsSpecific adsorbed anions
−
Stern-layer
+Volume water
Metal ε = ∞
Metal Ψs
representation of the solid-liquid interface [92].
lculate the electrical potential, Ψ, near charged interfaces.
s considered with a homogenous distributed electrical charge
ntact with a liquid. Generally charge density and potential are
ation [88-90]:
0εεeρ
2 −= (3.49)
the potential distribution can be calculated once the position
The complication in our case is that the ions in solution are
distributions, and thus the charge distribution in the liquid, is
cannot be found only by applying the Poisson equation.
required.
mula is the Boltzmann equation. If we have to bring an ion in
oser to the surface, electric work Wi has to be done. The local
(3.50)
ion sort in the volume phase, given in particles/m3. The local
ds on the electrical potential at the respective place. For
at a certain place in the solution is positive, then at this place
while the cation concentration is reduced.
ε = 78
Secondary bond water ε ≈ 32 water ε ≈ 6
Theory27
Now it is assumed that only electrical work has to be done. It is furthermore
neglected for instance that the ion must displace other molecules. In addition, it is
assumed that only a 1:1 salt is dissolved in the liquid. The electrical work required to
bring a charged cation to a place with potential Ψ is W + = qΨ. For an anion it is W − =
− qΨ. The local anion and cation concentration n− and n+ are related with the local
potential Ψ through the Boltzmann factor:
T/kqe n-n B
0Ψ
= , T/kq-
e nn B0
Ψ=+ (3.51)
n0 is the volume concentration of the salt. The local charge density is:
ρe = q(n− − n+) = n0 q
−−
TkqΨ
eTk
qΨ
e BB (3.52)
Substituting the charge density into Poisson eduation gives the Poisson−Boltzmann
equation:
−
−=∇Tkz)y,(x,qΨ
eTkz)y,(x,qΨ
e εε
qnΨ2 BB
0
0 (3.53)
This is a partial differential equation of second order. In most cases, it cannot be solved
analytically. Nevertheless, some simple cases can be treated analytically.
One dimensional geometry
A simple case is the one-dimensional situation of a planar, infinitely extended
plane. In this case the Poisson-Boltzmann equation only contains the coordinate vertical
to the plane:
−
−=Tk
(x)qΨ
eTk
(x)qΨ
e εε
q n2dx
Ψ2d BB
0
0 (3.54)
before it is solved this equation for the general case, it is illustrative to treat a special
case:
Theory28
A. Low potential
How does the potential change with distance for small surface potential? “Small” means,
strictly speaking q|Ψ0| << kBT. At room temperature that would be ≈ 25 mV. Often the
result is valid even for higher potentials, up to approximately 50-80 mV. With small
potentials it can be expanded that the exponential functions into a series and neglect all
but the first (i.e. the linear) term:
Ψ Tkεε
2q 2n....
TkqΨ1
TkqΨ1
εεq n
2dx
Ψ2d
B
0
0
BB0
0 =
±+−+≈ (3.55)
This is some times called the linearized Poisson-Boltzmann equation. The general
solution of the linearized Poisson-Boltzmann equation is:
Ψ(x) = C1 kxe− + C2 kxe (3.56)
with
Tkεε
2q2nk
B
0
0= (3.57)
C1 and C2 are constants which are defined by boundary conditions. For a simple double
layer the boundary conditions are Ψ (x → ∞) =0 and Ψ (x = 0) = Ψ 0. The first boundary
condition guarantees that with very large distances the potential disappears and does not
grow infinitely. From this follows C2 = 0. From the second boundary condition follows
C1 = Ψ 0. Hence, the potential is given by:
Ψ = Ψ 0 kxe− (3.58)
The potential decreases exponentially. The typical decay length is given by λD = k−1. It
is called the Debye length.
The Debye length decreases with increasing salt concentration. That is intuitively clear:
The more ions are in the solution, the more effective is the shielding of the surface
charge. If one quantifies all the factors for water at room temperature, then for a
monovalent salt with concentration c the Debye length is λD =3/ c Å, with c in mol/l.
Theory29
B. Arbitrary potential
Now comes the general solution of the one-dimensional Poisson-Boltzmann
equation. It is convenient to treat the equation with dimensionless potential y ≡ qΨ/kT.
The Poisson-Boltzmann equation becomes thereby:
sinhy2ky-e-ye21.
Tk εε
2q 2ny-e-ye Tkεε
q n2dx
y2d
B0
0
B 0
0 =
=
≈ (3.59)
To obtain this it is used:
2dx
2d Tk
q2dx
y2d
B
Ψ≈ and sinh y = 1/2 ( ye − ye− )
The solution of the differential equation 3.59 is:
Ckxy/2e
y/2e +−=
+
−
1
1ln (3.60)
The potential must correspond to the surface potential for x = 0, that means y(x = 0) =
y0. With the boundary condition one gets the integration constant
C/2ye
/2 ye
0
0=
+
−
1
1ln (3.61)
substitution results in
11
11
11
11ln1
1ln1
1ln
−++
++−=⇒
−=
−++
++−=
+
−−
+
−
/2 yey/2e
/2 yey/2ekx-e
kx/2 yey/2e
/2 yey/2e
/2 ye
/2 ye
y/2e
y/2e
0
0
0
0
0
0
(3.62)
solving the Eq. 3.62 for y/2e leads to the alternative expression:
kx-e/2 y
e/2y
e
kx-e/2y
e/2 y
ey/2e0
0
00
)1(1
)1(1
−−+
−++= (3.63)
The potential and the ion concentrations are shown as an example in the
illustration. A surface potential of 50 mV and a salt concentration (monovalent) of 0.1 M
were assumed.
Theory30
Figure 3.4.3.: The relation between the surface potential and salt concentrations.
It is clear that:
• The potential decreases approximately exponentially with increasing distance.
• The salt concentration of the counter ions decreases more rapidly than the
potential.
• The total ion concentration close to the surface is increased.
In the following illustration the potential, which is calculated with the linearized
form of the Poisson-Boltzmann equation (dashed), is compared with the potential which
results from the complete expression. It can be seen that the decrease of the potential
becomes steeper with increasing salt concentration. This reflects that the Debye length
decreases with increasing salt concentration.
Figure 3.4.4.: The relation between the surface potential and electrical double layer
thickness at different salt concentrations.
05
101520253035404550
0 0.5 1 1.5 2 2.5 3 3.5 4
Distance [nm]
Pote
ntia
l [m
V]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Con
cent
ratio
n [M
]Potential
Conc. Counter-ions
Conc. Co-ions
0
10
20
30
40
50
60
70
80
0 2 4 6 8 10
Distance [nm]
Pote
ntia
l [m
V]
fulllinearized
1 mM10 mM
0.1 M
Theory31
3.4.3.2.3.4.3.2.3.4.3.2.3.4.3.2. The The The The Grahame equationGrahame equationGrahame equationGrahame equation
How are surface charge and surface potential related? This question is important, since
the relation can be examined with the help of a capacity measurement. Theoretically this
relation is described by the Grahame equation. One can deduce the equation easily from
the so called electroneutrality condition. This condition demands that the total charge,
i.e. the surface charge plus the charge of the ion in the whole double layer, must be zero.
The total charge in the double layer is ∫∞
0dx eρ and it is leads to:
0xdxdΨεε
0-dx 2dx
Ψ2dεε0
dx eρσ00 =
∫∞
==∫∞
−= (3.64)
in the final step we has used dΨ/dx x = ∞ = 0. With
dxdΨ
Tkq
dxT)Ψ/k d(q
dxdy
B
B == and 2y sinh2k
dxdy −= follows
=
T2kqΨ
sinhTkεεn 8σB
0B 0 0 (3.65)
For small potential one can expand sinh into a series (sinh x = x + x3/3! + ...) and break
off after the first term. That leads to the simple relation:
D
00λΨεε
σ = (3.66)
The following illustration shows the calculated relation between surface tension and
surface charge for different concentrations of a monovalent salt.
Figure 3.4.5.: The relation between the surface potential and surface charge density at
different salt concentrations.
0
20
40
60
80
100
120
140
0 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2
Surf
ace
pote
ntia
l [ mV]
Surface charge density [e/nm2]
Theory32
It is clear that:
• For small potential σ is proportional to Ψ0.
• For high potential σ rises more steeply then Ψ0.
• The capacity dσ/dΨ0 grows with increasing surface potential.
• Depending on the salt concentration, the linear approximation (dashed) is valid
till Ψ0 ≈ 40....80 mV.
3.4.3.3.3.4.3.3.3.4.3.3.3.4.3.3. The capacity of the double layer The capacity of the double layer The capacity of the double layer The capacity of the double layer
A plate capacitor has the capacity:
C = dQ/dU = εε0 A/d (3.67)
Q: Charge, U: Applied voltage, A Area, d: Distance. The capacity of the electrical
double layer per area is thus
=
==
T2kqΨ
cosh λεε
T2kqΨ
coshTkεεn 22q
dΨdσC
B
0
D
0
B
0
B
00
0 (3.68)
it can be expanded cosh into a series (cosh x = 1 + x2/2! + x4/4! + ...) and for small
potential it can be broken off after the second term, then results:
D
0λεε
C = (3.69)
3.4.4.3.4.4.3.4.4.3.4.4. Additional description of the Additional description of the Additional description of the Additional description of the electrical double layerelectrical double layerelectrical double layerelectrical double layer
The Gouy-Chapman model is unrealistic for high surface potential Ψ0 and small
distances because then the ion concentration becomes too high. Stern tried to handle this
problem by dividing the region close to the surface into two regions:
• The Stern-layer, consisting of a layer of ions, which as assumed by Helmholtz,
are directly adsorbed to the surface.
Theory33
• A diffuse Gouy-Chapman layer.
In reality all models can only describe certain aspects of the electrical double
layer. The real situation at a metal surface might for instance look like in the following
illustration.
Figure 3.4.3.: Schematic represantation of the double layer, Stern layer and diffusion
layer (GC layer)
The surface potential, Ψ0, is determined by the external potential and by
adsorbed ions. These ions are not distinguishable from the metal itself (e.g. Ag+ or I− on
AgI). Next comes a layer or relatively tightly bound, hydrated ions (e.g. H+ or OH− on
oxides or proteins). If this layer exists it contributed to Ψ0. The so called Inner
Helmholtz Plane, I.H.P., marks the centre of these ions. The Stern layer consists of
adsorbed, hydrated ions. The Outer Helmholtz Plane, O.H.P., goes through the centre of
these ions. Finally there is the diffuse layer. The potential at the distance where it
originates corresponds roughly to the zeta-potential.
−
−
Stern−layer
GC−layer (diffuse−layer)
Ψd
Ψo
Pote
ntia
l [m
V]
Distance [nm]
Experimental work34
4. Experimental work4. Experimental work4. Experimental work4. Experimental work
4.1.4.1.4.1.4.1. Fluidized bed experimentsFluidized bed experimentsFluidized bed experimentsFluidized bed experiments
4.1.1. 4.1.1. 4.1.1. 4.1.1. Fluidized bed measurement equipmentFluidized bed measurement equipmentFluidized bed measurement equipmentFluidized bed measurement equipment
The fluidized bed crystallization system, already used and discussed e.g. by [1, 4,
62, 101] is consists of a pump and two heat exchangers in addition to cell. The fluidized
bed cell is shown in Fig. 4.1., and is made of acrylicglass. The crystallization system is
assembled in the process of chemical engineering department’s laboratory. The pump is
a centrifugal pump (Iwaki magnet pump, model MD 30R-220N), the two heat
exchangers were connected to two water baths, one (HAAKE N3, typ 001-5722)
adjusted to keep the solution undersaturation in the vessel (heating) while the other
(HAAKE F3, typ 002-0991) operates as a cooler to create the required supersaturation
and undersaturation level in the growth and dissolution zone, each water bath is
6.6.6.6. Interpretation of resultsInterpretation of resultsInterpretation of resultsInterpretation of results
6.1.6.1.6.1.6.1. The The The The magnitude of the two resistancesmagnitude of the two resistancesmagnitude of the two resistancesmagnitude of the two resistances (diffusion and integration steps) (diffusion and integration steps) (diffusion and integration steps) (diffusion and integration steps)
The overall crystal growth phenomenon can be considered as the combination of
3 steps, namely volume diffusion, surface reaction, and heat transfer. The first step is the
mass transport of the growth units by diffusion or convection from the bulk of the
solution to the crystal surface. The second step is the integration of units into the crystal
lattice. The third step are heat related effects from the liberation of the crystallization
heat when the crystal grow, and from the heat transfer connected with the mass transfer
from and to the phase boundary (liquid/solid) [1−3]. Here the effect of heat transfer can
be neglected due to the fact that the value of βd <10 −2 [3], and the value of −∆H/(1−wb)
is less than 700 J g−1[113]. Therefore, the crystal growth processes from solutions are
dominantly either diffusion or reaction controlled. A change in the dominating growth
mechanism arises by a change of temperature level or e.g. by traces of certain impurities
In the aqueous electrolyte solution two different species can be found, the
dissociated ions (cations and anions) and the water molecules, there are different types
of forces in the system: water-water, ion-ion, and water-ion interaction [118]. The
simplest reaction of the metals, M n+, in aqueous solutions is the loss of a proton to
give the hydroxy species, M(OH) (n-1)+. The coordination of metal ion to a water
molecule will make a proton loss easier. The greater the positive charge on the metal
ion, the easier it should be for proton to dissociate from an attached water molecule.
Such an equilibrium is given the following equation [119]:
M n+ + 2H2O = M(OH) (n−1)+ + H3O + (6.1)
6.3.1.6.3.1.6.3.1.6.3.1. The effect of pH The effect of pH The effect of pH The effect of pH (MgSO(MgSO(MgSO(MgSO4444••••7H7H7H7H2222O experiments)O experiments)O experiments)O experiments)
The dissociation of a proton from water molecules may effect the structure of
the solution. By addition of Mg+2 ions to the water, the hydronium ions are generated
as a result of the hydration of Mg+2 according to the equilibrium in Eq. 6.1 and as
shown in the following expression:
Mg +2 + 2H2O = Mg(OH) + + H3O + (6.2)
The addition of hydronium ions (H3O+) to the solution will affect the position
of the equilibrium. According to Le Chatelier principle the position of chemical
equilibrium always shifts in a direction that tends to relieve the effect of an applied
stress [120]. Thus, an increase H3O+ ions in the solution causes to shift the position of
the equilibrium to the left side. i.e. the solubility of the salt will decrease. While, the
addition of OH− ions will have the opposite effect. This may be accepted as an
explanation for the change in the solubility of the solution after the change in the pH-
value of the solution.
The change in the structure of the solution by the presence of various ions can
be estimated from the entropy data (order or disorder of the system) shown in Table
6.3.1.. If the entropy of an electrolyte has a more negative value, then it is a structure
Interpretation of results59
former. If the value is less negative, then it is a structure breaker. For this work the
following can be summarised [121]:
• H3O + is a slight structure breaker.
• Cations smaller or more highly charged than H3O + are structure formers.
• OH− is a structure breaker, while SO4−2 is a structure former.
Table 6.3.1.: Physical properties of cations and anions at 298 K [122].
Physical properties
Ion Ionic Radii
[A°]
Hhyd
[KJ.mol−1]
Shyd
[KJ.mol−1.K−1]
Ghyd
[KJ.mol−1]
H3O+ −− −1129 −131 −1090
Na + 0.95 −444 −110 −411
K+ 1.33 −360 −74 −338
Fe +2 0.76 −2305 −383 −2191
Mg +2 0.65 −1999 −311 −1906
Ni +2 0.72 −2490 −396 −2372
Pb +2 1.2 −1785 −228 −1717
Cl− 1.81 −340 −76 −340
OH− 1.4 −423 −149 −379
SO4 −2 1.5 −1145 −263 −−
In supersaturated solution, neutral solution, Mg +2 ions will move towards the
crystal surface (the crystal growth will be enhanced), while in undersaturated solution
Mg +2 ions will leave the crystal surface towards the solution (the dissolution rate will
be enhanced). In the same way, the addition H3O + ions to the solution (acidic
medium) will make the solution unstable (structure breaker). Consequently, in
supersaturated solution it is proposed that, the Mg +2 ions prefer to remain in the
solution (as aqueous ions Mg(OH)+ see Eq. 6.2). Therefore, the number of Mg+2 ions
arriving to the crystal surface will be reduced, therefore, the growth rate will be
Interpretation of results60
suppressed. In undersaturated solution the number of Mg+2 ions that must leave the
crystal surface towards the solution will be reduced, therefore, the dissolution rate will
be suppressed. This may be accepted as an explanation for a lower growth and
dissolution rate of MgSO4•7H2O crystals in acidic/alkaline solutions compared with
the neutral solutions.
6.3.2.6.3.2.6.3.2.6.3.2. The effect of The effect of The effect of The effect of K K K K ++++ ions ions ions ions ((((MgSOMgSOMgSOMgSO4444••••7H7H7H7H2222O experiments)O experiments)O experiments)O experiments)
It can also be seen from the thermodynamic properties of the ions shown in
Table 6.3.1.. The smaller the ion is, the stronger the interaction (higher hydration
energy) between the ion and water molecules in the coordination sphere, that is to say
univalent, divalent and trivalent positive ions (cations) are hydrated. The largest
enthalpy of hydration can be expected for the smallest ion (e.g. Mg+2 has enthalpy –
1999 kJ mol-1). Therefore, the presence of such an ion in the solution has more
tendencies for interaction with water molecules than the largest ion, especially, when
the latter has such a relatively small enthalpy of hydration (K+ has enthalpy –360 kJ
mol –1). It is proposed that such an interaction stands behind the increase in the
solubility of MgSO4•7H2O in the presence of K+ ions in the solution. The later
inference has proven the results concerning the increase in the solubility of
MgSO4•7H2O by adding K2SO4 and KH2PO4 as impurities (see Table 5.2.1.).
As mentioned previously the change in the structure of the solution by the
presence of various ions can be estimated from the entropy data (order or disorder of
the system) shown in Table 6.3.1.. Here the following can be summarised:
• K + is a slight structure breaker.
• A cation smaller or more highly charged than K+ is a structure former.
Here it is proposed, that the presence of K+ ions in the solution will make the solution
unstable (structure breaker). Consequently, K+ ions prefer to move toward the crystal
surface rather than to remain in the solution (as aqueous ions). Therefore, the number
of K+ ions adsorbing on the crystal surface will be increased by increasing the amount
of K+ ions adding to the solution, hence the number of Mg+2 ions arriving at the
crystal surface will be reduced (the growth rate will be suppressed). This may be
accepted as an explanation for suppressed growth rate compared to the pure solution.
Interpretation of results61
6.3.3.6.3.3.6.3.3.6.3.3. The effect of hydro-complex ions The effect of hydro-complex ions The effect of hydro-complex ions The effect of hydro-complex ions
Die Hauptzielsetzung der Untersuchung ist es, eine neue Erklärung für den Effektder Verunreinigungen oder des pH-Wertes der Lösung auf die Kinetik der Kristallisationvorzuschlagen. Die Hypothese, basiert darauf, dass die Kristallwachstumrate der hier nurbetrachteten anorganischen Salze aus wässeriger Lösungen durch dieOberflächenpotentialverteilung beherrscht wird.
In dieser Arbeit wird ein Wirbelbettkristallisator eingesetzt, um die Wachstums-
und Auflösungsgeshwindigkeit von MgSO4•7H2O und NaCl Kristallen zu vermessen. Inden Experimenten wurden die Übersättigung, die Konzentration der Verunreinigungenund die pH-Werte in der Lösung verändert. Die Wachstumsraten wurden inAbhängigkeit des Übersättigungniveaus ermittelt. Die Exponenten und die Konstantender Wachstumskinetik wurden experimentell ermittelt. Der Wirkungsgrade wurde ausden Wachstumsratedaten erhalten. Die relative Größe der zwei in Reihe, geschaltetenWiderstände der Diffusion und der Integration wurde so abgeschätzt. Die Auswertungdes Wirkungsgrades zeigt folgendes:
1. In Abwesenheit von Verunreinigungen ist die Kristallwachstumsgeshwindigkeit vonNaCl diffusionskontrolliert. Die Verunreinigungen rufen jedoch eine Veränderungdes Wachstumsmechanismus hervor, d.h. die Gegenwart der Verunreinigungen führtzu einer wichtigeren Rolle des Integrationsschrittes.
2. Zum Kontrollmechanismus des Kristallwachstums von MgSO4•7H2O in reiner alsauch in unreiner Lösung tragen der Diffusions- und der Intigrationsschritt bei. ImFalle der reinen Lösung liefert der Diffusionsschritt gegenüber demGesamtkristallwachstums einen stärkeren Widerstand als der Intgrationsschritt. DieVerunreinigungen lassen dagegen die Rolle des Integrationsschrittes dominieren.
Folglich kann der Effekt der unterschiedlichen Verunreinigungen auf die
Wachstumsrate von MgSO4•7H2O und NaCl Kristallen in drei Gruppe geteilt werden:
1. Thermodynamische Effekte: Verunreinigungen, welche die Gleichgewichtsättigungs-konzentration verändern.
2. Kinetische Effekte: Verunreinigungen, welche die Kristallwachstumrate von
MgSO4•7H2O und NaCl Kristallen verringern.
Zusammenfassung85
3. Thermodynamische Effekte sowie kinetische Effekte: Verunreinigungen, welche die
Löslichkeit sowie die Wachstumsrate von MgSO4•7H2O und NaCl der Kristallebeeinflussen.
Die von mir ermittelten Daten der Kristallwachstumsraten in wässrigenLösungen in Abhängigkeit von den Konzentrationen der Verunreinigungen wurdensowohl nach dem Gesichtspunkt von Cabrera und Vermileya [14] als auch nach Kubota
und Mullin [49] diskutiert. Der Wert des Verunreinigungseffektes, αθeq, der aus derAnalyse der Daten aus der Wachstumskinetik ermittelt wurde, steht in guterÜbereinstimmung mit dem berechneten Wert aus den direktenAdsorptionsexperimenten. Die Werte des mittleren Raumes zwischen benachbartenadsorptionsaktiven Stellen und des mittleren Abstandes zwischen benachbarten mitVerunreinigung adsorpierten Stellen wurden verglichen.
Zuletzt wurde über elektrophoretische Mobilitätsuntersuchungen mittels der
Laser-Doppler-Elektrophorese (ζ-Potentialmessungen) für MgSO4•7H2O Kristalleberichtet. Diese Untersuchungen bei einem anorganischen Salz wurden zum ersten Mal
überhaupt durchgeführt und erlauben es, die Oberflächenladung der MgSO4•7H2OKristalle in ihren gesättigten Lösungen vorauszusagen. Folglich kann die Kenntnis des
Oberflächenpotentials mittels Messung des ζ-Potentials helfen, dieKristallisationsphenomene zu erklären, die bis jetzt unbekannt sind.
Im allgemeinen zeigen die Ergebnisse, dass die MgSO4•7H2O Kristalle eine
positive ζ-Potentials Ladung besitzen. Bei niedrigen pH-Werten erhält die Oberflächeeine positivere Ladung, während bei hohen pH-Werten der Aufbau einer negativenLadung stattfindet, folglich wird das Kristallwachtum unterdrückt. In dieser Studie
wurde nachgewiesen, dass die Kristallwachstumsgeschwindigkeit von MgSO4•7H2Odurch Spuren von Fe+2 bzw. Ni+2 Ionen unterdrückt werden. Hierbei kam dasWachstumsverhalten angemessen mittels eines Mechanismus erklärt werden, bei demein adsorbiertes Hydrolyseprodukt (eine angenommene aktive Spezies) von denhydratisierten Fe+2 bzw. Ni+2 Ionen die Wachstumsgeschwindigkeit unterdrückt. DieOberflächenladung wurde bei der Zugabe von Fe+2 und Ni+2 verändert. Dieses Ergebnislegt den Schluss nahe, dass eine Adsorption von komplexen Ionen an die
Kristalloberflache möglich ist. Folglich verursacht die Änderung des Wertes von ζ oder
dessen Vorzeichen eine Unterdrückung des Wachstums von MgSO4•7H2O.
Notation86
9. Notation9. Notation9. Notation9. Notation
A Hydro-complex compound of divalent ions, inactive both
for adsorption and growth suppression.
A1* First hydrolysis product of divalent ions, active both for
adsorption and growth suppression.
A2* Second hydrolysis product divalent ions, active only for
adsorption.
[A] [mol/dm3] Concentration of A.
[A1*] [mol/dm3] Concentration of A1
*.
[A2*] [mol/dm3] Concentration of A2
*.
a [m2] The length of growth unit.
al, as [M] Activity in the solution and at the crystal surface.
C [F cm−2] Capacity of the electrical double layer per area.
C* [kg/m3] Saturation concentration.
Cb [kg/m3] Bulk concentration.
Ci [kg/m3] Interface concentration.
∆C [kgsalt/m3soln] Supersaturation.
c [M] Ion concentration.
cimp [ppm or mol. fra.] Impurity concentration.
cp [J kg−1 K−1] Specific heat capacity.
Da [-] Damköhler number.
E [V/m] Electric field strength.
Emig [J] Activation energy of migration.
Ff [N] Frictional force.
f [kg/s] Friction coefficient.
f1(pH) [-] Fraction of A1* present in the solution (Eq. 6.9).
G [m/s] Overall linear growth rate.
g [-] Order of growth rate.
Ghyd [KJ mol−1] Hydration energy.
h [J m−2 s−1 K−1] Heat transfer coefficient.
Hhyd [KJ mol−1] Hydration enthalpy.
∆H [J/kg] Heat of crystallization.
K [(mol/mol)−1] Langmuir constant.
k [m−1] Reverse length.
kB [J K−1] Boltzmann constant.
Notation87
kd [m/s] Dissolution rate constant.
kG [kg/m2s(kg/m3sol) – g] Overall growth rate constant.
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(Dr. in Chemical Engineering) at the Martin-Luther-Universität Halle-Wittenberg, Department of Process Chemical Engineering in 19.12.2002. Ph.D. thesis entitle: Effects of Additives in Solution Crystallization. (M.Sc. in Chemical Engineering) at the University of Jordan, College of Engineering/ Department of Chemical Engineering in 20.06.1999.
GPA (3.5 from 4).
M.Sc. thesis entitle: Measurement of Growth Rate and Dissolution Rate of Potassium Dihydrogen Phosphate (KH2PO4) Crystals.
(B.Sc. in Chemical Engineering) at the University of Baghdad, College of Engineering, Department of Chemical Engineering in 01.07.1995.
Worked as a research assistant at the University of Jordan, College of Engineering, Chemical Engineering Laboratory with main focus on industrial crystallization process. Worked as a lectures assistant teaching courses in Thermodynamic, Mass transfer, and Reaction engineering; at the University of Jordan College of Engineering/ Department of Chemical Engineering. Worked as a computer Lab. technician with knowledge of (AutoCAD, HTML Design for Web Pages, Win95, Win98, and Win3.11. and application. Such as “WinWord, Excel, PowerPoint”, and MATLAB); at the University of Jordan, College of Engineering, Computer Laboratory. Worked as a research assistant in design and simulation of chemical engineering at the University of Jordan, College of Engineering, Department of Chemical Engineering.